Applications of Fibonacci Numbers

Applications of Fibonacci Numbers

Volume 8

Proceedings of 'The Eighth International Research Conference on Fibonacci Numbers and Their Applications',

Rochester Institute of Technology, Rochester, New York, U.S.A., June 22-26, 1998

edited by

Fredric T. Howard Department of Mathematics and Computer Science,

Wake Forest University, Winston-Salem, North Carolina, U.S.A.

SPRINGER SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-94-010-5851-3 ISBN 978-94-011-4271-7 (eBook) DOI 10.1007/978-94-011-4271-7

Cover figure by John C. Turner

Printed on acid-free paper

All Rights Reserved © 1999 Springer Science+Business Media Dordrecht

Originally published by Kluwer Academic Publishers in 1999 Softcover reprint of the hardcover 1 st edition 1999

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical,

including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

TABLE OF CONTENTS

A REPORT ON THE EIGHTH INTERNATIONAL CONFERENCE .................................... vii LIST OF CONTRIBUTORS TO THIS PROCEEDINGS ......................................................... ix FOREWORD ........................................................................................................................ xvii THE ORGANIZING COMMITTEES .................................................................................... xix LIST OF CONTRIBUTORS TO THE CONFERENCE ........................................................ xxi INTRODUCTION .............................................................................................................. xxiii

ON THE GENERALIZED BINOMIAL COEFFICIENTS DEFINED BY STRONG DIVISIBILITY SEQUENCES

Shiro Ando and Daihachiro Sato ................................................................................... l ON TRIANGLES AND SQUARES MARKED WITH GOLD POINTS-STUDIES OF GOLDEN TILES

Vassia K. Anatassova and J.C. Turner ....................................................................... 11 MULTIVARIATE PASCAL POLYNOMIALS OF ORDER K WITH PROBABILITY APPLICATIONS

Demetrios L. Antzoulakos and Andreas N. Philippou ................................................. 27 FIBONACCI PLANES AND SPACES

K.T. Atanassov and A.G. Shannon ............................................................................ 43 THE SMALLEST POSITIVE INTEGER HAVING Fk REPRESENTATIONS AS SUMS OF DISTINCT FIBONACCI NUMBERS

Marjorie Bicknell-Johnson ......................................................................................... 47 THE ZECKENDORF-WYTHOFF ARRAY APPLIED TO COUNTING THE NUMBER OF REPRESENTATIONS OF N AS SUMS OF DISTINCT FIBONACCI NUMBERS

Marjorie Bicknell-Johnson ......................................................................................... 53 COMPOSING WITH SEQUENCES: •.• BUT IS IT ART?

John A. Biles ............................................................................................................ 61 INVARIANTS FOR LINEAR RECURRENCES

Mihai Caragiu and William Webb .............................................................................. 75 BASE 10 RATS CYCLES AND ARBITRARILY LONG BASE 10 RATS CYCLES

Curtis Cooper and Robert E. Kennedy ....................................................................... 83 QUINTICS x5 - 5x - k, THE GOLDEN SECTION, AND SQUARE LUCAS NUMBERS

Michele Elia and Piero Filipponi ............................................................................... 95 THE PASCAL-DE MOIVRE MOMENTS AND THEIR GENERATING FUNCTIONS

Larry Ericksen ......................................................................................................... 103 INVESTIGATING SPECIAL BINARY SEQUENCES WITH SOME COMPUTER HELP

Daniel C. Fielder and Cecil O. Alford ...................................................................... 121 INTEGRATION SEQUENCES OF JACOBSTHAL AND JACOBSTHAL-LUCAS POLYNOMIALS

Piero Filipponi and Alwyn F. Horadam .................................................................... 129

v

vi TABLE OF CONTENTS

A PROPERTY OF THE UNIT DIGITS OF RECURSIVE SEQUENCES Herta T. Freitag, Marjorie Bicknell-Johnson and George M. Phillips ........................ 141

ON GENERAL DIVISIBILITY OF SUMS OF INTEGRAL POWERS OF THE GOLDEN RATIO

Herta T. Freitag and Daniel C. Fielder .................................................................... 149 SYLVESTER'S ALGORITHM AND FIBONACCI NUMBERS

Herta T. Freitag and George M. Phillips ...................................•.........•.................... 155 ON THE CHARACTERISTIC POLYNOMIAL OF THE J-TH ORDER FIBONACCI SEQUENCE

George W. Grossman and Sivaram K. Narayan ......•..........•.......•...•.............••........... 165 QUASI MORGAN-VOYCE POLYNOMIALS AND PELL CONVOLUTIONS

A.F. Horadam .................•..........•..........•.........•.........•.......•......•...•..............•............ 179 ON AN ASYMPTOTIC MAXIMALITY OF THE FIBONACCI TREE

Yasuichi Horibe ....................•...................•..............•..•...............•..•...•................•..... 195 GENERALIZATIONS OF A FIBONACCI IDENTITY

F.T. Howard .•...•...................................................................................................... 201 SOME GENERALIZATIONS OF WOLSTENHOLME'S THEOREM

William A. Kimball and William A. Webb ••.......................................•...................... 213 CARD SORTING RELATED TO FIBONACCI NUMBERS

Clark Kimberling •..........•..................•••..................................................................... 219 ON THE INHOMOGENEOUS GEOMETRIC LINE-SEQUENCE

Jack Lee ....••.........................•...........................................................•....................... 233 FIBONACCI NUMBERS OF THE FORM k2 + k + 2

Florian Luca ..........•................................................•..................•.............................. 241 ON CERTAIN POLYNOMIALS OF EVEN SUBSCRIPTED LUCAS NUMBERS

R.S. Melham ...................................................................................•......................... 251 ON THE RANK OF APPEARANCE OF LUCAS SEQUENCES

Siguna Muller ........................................................................................................... 259 ALGORITHMIC SIMPLIFICATION OF RECIPROCAL SUMS

Stanley Rabinowitz .•.........•..•..................••..................•...................•.......•..•.......•....... 277 SOLVED AND UNSOLVED PROBLEMS ON PSEUDOPRIME NUMBERS AND THEIR GENERALIZATIONS

Andrzej Rotkiewicz .........•..•...•...•.......•..••....•...•......•..••.....•........•..•...............••..•........ 293 SOME RELATIONSHIPS AMONG VIETA, MORGAN-VOYCE AND JACOBSTHAL POLYNOMIALS

A.G. Shannon and A.F. Horadam ......................................................•...................... 307 SPECIAL MULTIPLIERS OF LUCAS SEQUENCES MODULO pr

Lawrence Somer ..............................................................................••..........•.........•.. 325 DIGITAL HALFTONING USING ERROR DIFFUSION AND LINEAR PIXEL SHUFFLING

John Szybist and Peter G. Anderson .•....................................................................... 337 ON VECTOR SEQUENCE RECURRENCE EQUATIONS IN FIBONACCI VECTOR GEOMETRY

John C. Turner ..................•..•...........•...•...................•.............................................. 353 CONSTRUCTING IDENTITIES INVOLVING Kth-ORDER F-L NUMBERS BY USING THE CHARACTERISTIC POLYNOMIAL

Chizhong Zhou .••................•..••.................................................................................. 369 SUBJECT INDEX ..................................................................•............................................. 381

A REPORT ON THE EIGHTH INTERNATIONAL CONFERENCE

ON FIBONACCI NUMBERS AND THEIR APPLICATIONS

Rochester Institute of Technology, which has been internationally respected as a world leader in career-oriented and professional education since 1829, was the inspired choice of setting for our Eighth Conference. We are indeed grateful to Rochester Institute of Technology, as well as to the Fibonacci Association, for sponsoring our conference. A special word of thanks goes to Dr. Albert Simone, President of Rochester Institute of Technology, Wiley R. McKinzie, Dean of Applied Science and Technology, and Dr. Walter A. Wolf, Chair of The Computer Science Department.

The participants came from eighteen different countries: 28 from the USA, three each from Australia, Canada and Japan, two each from Germany and Italy, and one participant from each of Austria, Belarus, Cyprus, Denmark, France, Greece, Iceland, New Zealand, Poland, Romania, and Scotland. Five of the presenters were women. There were four mathematicians who have attended all of the eight conferences, many who have attended several and, most happily, several who have attended for the first time. The mltgnetism of Fibonacci-type mathematics drew even some who did not present a paper. The ages ranged from a few who were in their twenties to one who will soon earn the title of nonagenarian.

There have been two major changes since our last conference, both involving Jerry Bergum. After eighteen years as editor of the Fibonacci Quarterly, he has handed over the baton to Curtis Cooper. We wish Curtis all success and fulfillment in his new role. At the same time, Jerry has been succeeded as conference organizer by Fred Howard, who also has our best wishes. Fred is already widely respected for his wisdom and kindliness.

We hope that Jerry will attend many more of our conferences. We deeply appreciate all he has done. He has been in a very real sense the heart and soul of our Association. We would also like to renew our thanks to Calvin Long for his continuing work as our President. Our discussions have been illuminated by his fine mathematical insights.

This was a conference where all of the talks were attended by almost all of the participants, who appreciated the diversity of topics covered and the remarkable level of quality of the papers. All of the presentations displayed the high enthusiasm of the speakers for their studies, and they all showed enjoyment over the opportunity of sharing their ideas with each other.

As well as working hard (41 talks in five days!), the group also enjoyed some delightful social events, the highlight being a "cook-out" at the Andersons' home with Peter and Jane, our gracious hosts. Through his wit and warmth, Peter immediately set the stage for a conference where we not only saw a fellow-mathematician in each other (which would already be enjoyable) but, moreover, a friend. We are deeply grateful to Peter and his helpers for all their hard work in preparing those delightful outings for us, and the extra care they took in looking after us in Rochester.

vii

viii A REPORT ON···

The friendships created by this sequence of Fibonacci conferences has produced many worthwhile results in this area of mathematics. At this conference we have enjoyed renewing old friendships and beginning new ones. "The Goddess Mathesis" (to use Howard Eves' term) looks favorably on these friendships.

Finally, we had to part. But now we greatly look forward to meeting again in two years: in Luxembourg in 2000.

H ena T. Freitag

LIST OF CONTRIBUTORS TO THIS PROCEEDINGS*

Professor Arnold Adelberg Grinnell College Department of Mathematics & Computer Science P.O. Box 805 Grinnell, Iowa 50112-0806

Professor Octavian Agratini Babes-Bolyai University Faculty of Mathematics and Informatics str. Kogalniceann Nr. 1,3400 Cluj-Napoca, Romania

Professor Cecil O. Alford (pp. 121-128) School of Electrical and Computer Engineering Georgia Institute of Technology Atlanta, GA 30332-0250

Professor Peter G. Anderson (pp. 337-352) Rochester Institute of Technology Department of Computer Science 102 Lomb Memorial Drive Rochester, NY 14623

Shiro Ando (pp. 1-10) 5-29-10 Honda Kokubunji-shi Tokyo 185-0011 JAPAN

Professor Angel Andreu Department of Mathematics Monroe Community College 1000 East Henrietta Rd. Rochester, NY 14623

*This list includes all authors and coauthors of papers presented at the conference even if their paper was rejected, published elsewhere or not submitted to the proceedings. Those who attended but did not present a paper are also in this list.

ix

x

Professor Demetrios L. Antzoulakos (pp. 27-42) Department of Statistics and Actuarial Science University of Piraeus 18534 Piraeus Greece

Professor Vassia K. Anatassova (pp. 11-26) Centre of Biomedical Engineering Bulgarian Academy of Sciences P.O. Box 12, 113 Sofia Bulgaria

Professor K T. Atanassov (pp. 43-46) Centre of Biomedical Engineering Bulgarian Academy of Sciences Sofia-1113 Bulgaria

Professor Gerald E. Bergum Box 2201 Computer Science Department South Dakota State University Brookings, SD 57007-1596

CONTRIBUTORS TO THIS PROCEEDINGS

Dr. Marjorie Bicknell-Johnson (pp. 47-52; 53-60; 141-148) 665 Fairlane Avenue Santa Clara, CA 95051

Professor John A. Biles (pp. 61-74) Information Technology Department Rochester Institute of Technology 102 Lomb Memorial Drive Rochester, NY 14623-5608

Professor Mihai Caragiu (pp. 75-82) Department of Pure and Applied Mathematics Washington State University Pullman, WA 99164-3113

Professor Walter Carlip 18 Garfield Avenue Athens, OR 45707

Professor Charles Cook 1 Louise Circle Sumter, SC 29150

Professor Curtis Cooper (pp. 83-94) Department of Mathematics and Computer Science Central Missouri State University Warrensburg, MO 64093

CONTRIBUTORS TO THIS PROCEEDINGS

Professor Karl Dilcher Dept of Mathematics, Statistics & Computer Science Dalhousie University Halifax NS Canada B3H 3J5

Professor Ernest Eckert Sondervangsvej 43 9000 Aalborg Denmark

Professor Michele Elia (pp. 95-102) Dip. di Elettronica Politecnico di Torino C.so Duca degli Abruzzi 24 1-10129 Torino, Italy

Mr. Larry Ericksen (pp. 103-120) P.O. Box 172 Millville, NJ 08332

Professor Daniel C. Fielder (pp. 121-128; 149-154) School of Electrical and Computer Engineering Georgia Institute of Technology Atlanta, GA 30332-0250

Mr. Piero Filipponi (pp. 95-102; 129-140) Fondazione Ugo Bordoni Via B. Castiglione 59 1-00142 Roma, Italy

Ms. Herta T. Freitag (pp. 141-148; 149-154; 155-164) B40 Friendship Manor 320 Hirshberger Road Roanoke, Virginia 24012

Professor George W. Grossman (pp. 165-178) Department of Mathematics Central Michigan University Mt. Pleasant, MI 48859

Professor Helen Grundman Dept. of Mathematics Bryn Mawr College 101 North Merion Avenue Bryn Mawr, PA 19010-2899

Professor Heiko Harborth Technishce Universitat Braunschweig Germany

xi

xii

Professor Deborah L. Harrell Department of Mathematics Salem College 507 Wachovia St. Winston-Salem, NC 27101

Professor Evelyn Hart Colgate University Department of Mathematics 13 Oak Drive Hamilton, NY 13346-1398

CONTRIBUTORS TO THIS PROCEEDINGS

Professor A:F. Horadam (pp. 129-140; 179-194; 307-324) The University of New England Armidale Australia 2351

Professor Yasuichi Horibe (pp. 195-200) Department of Applied Mathematics Science University of Tokyo 1-3 Kagurazaka, Shinjuku-ku Tokyo 162-8601 Japan

Professor F. T. Howard (pp. 201-212) Mathematics and Computer Science Wake Forest University Winston-Salem, NC 27109

Kristin Halla Jonsdottir Sefgardar 28 170 Seltjarnarnes Iceland

Professor Robert E. Kennedy (pp. 83-94) Department of Mathematics and Computer Science Central Missouri State University Warrensburg, MO 64093

Professor William A. Kimball (pp. 213-218) University of Maryland University Park at Schwabisch Gmund Universitatspark 8, 73525 Schwabisch Gmund Germany

Professor Clark Kimberling (pp. 219-232) University of Evansville 1800 Lincoln Avenue Evansville, IN 47722

CONTRIBUTORS TO THIS PROCEEDINGS

Professor Ron Knott University of Surrey Department of Computing Guildford, Surrey GU2 5XH United Kingdom

Professor Harris Kwong State University of New York College at Fredonia Department of Mathematics and Computer Science Fredonia, NY 14063

Dr. Jack Lee (pp. 233-240) 280 86th Street Brooklyn, NY 11209

Professor Calvin Long Department of Mathematics Northern Arizona University Flagstaff, AZ 86011

Professor Florian Luca (pp. 241-250) Mathematics Department Bielefeld University Postfach 10 01 31 33 501 Bielefeld Germany

Dr. R.S. Melham (pp. 251-258) School of Mathematical Sciences University of Technology, Sydney PO Box 123, Broadway· NSW 2007 Australia

Mr. Mark D. Morgan 8911 Tamar Drive #302 Columbia, MD 21045

Professor Siguna Muller (pp. 259-276) Department of Mathematics University of Klagenfurt Universitatsstasse 65-67 A-9020 Klagenfurt, Austria

Professor Kenji Nagasaka Hosei University Koganei-Shi Tokyo, 184-8584 Japan

xiii

xiv

Professor Shigeru Nakamura Tokyo University of Mercantile Marine Etchujima Kotoku Tokyo Japan

Professor Sivaram K. Narayan (pp. 165-178) Department of Mathematics Central Michigan University Mt. Pleasant, MI 48859

Professor Richard Orr Rochester Institute of Technology Department of Mathematics Lomb Memorial Drive Rochester, NY 14623

Professor Andreas N. Philippou (pp. 27-42) Department of Mathematics University of Patras 26100 Patras Greece

CONTRIBUTORS TO THIS PROCEEDINGS

Professor George M. Phillips (pp. 141-148; 155-164) Mathematical Institute University of St. Andrews St. Andrews Scotland

Dr. Stanley Rabinowitz (pp. 277-292) 12 Vine Brook Road Westford, MA 01886-4212

Ms. Margaret Ribble 1007 Court Street Maryville, TN 37803

Professor J. Adair Robertson Peace College 15 East Peace St. Raleigh, NC 27604

Professor Andrzej Rotkiewicz (pp. 293-306) Institute of Mathematics Polish Academy of Sciences, ul. Sniadeckich 8 00-950 Warszawa, Poland

Professor Laura Sanchis Colgate University 13 Oak Drive Hamilton, NY 13346-1398

CONTRIBUTORS TO THIS PROCEEDINGS

Professor Daihachiro Sato (pp. 1-10) Department of Mathematics & Statistics University of Regina Regina, Saskatchewan, S4S OA2 Canada

Professor A.G. Shannon (pp. 43-46; 307-324) University of Technology Sydney, 2007 Australia

Professor Lawrence Somer (pp. 325-336) Department of Mathematics The Catholic University of America Washington, DC 20064

Mr. Colin Paul Spears 3407 Buena Vista Ave. Glendale, CA 91208

Professor John Szybist (pp. 337-352) Intrinsix Corporation 160 AlIens Creek Rd. Rochester, NY 14618

Professor Tomakazu Takahashi Hosei University, Graduate School Engineering Division, Systems Engineering 3-7-2, Kajino-cho, Koganei-shi Tokyo

Professor J.C. Turner (pp. 11-26; 353-368) Department of Mathematics and Statistics University of Waikato Private Bag 3105 Hamilton, New Zealand

Professor Theresa Vaughan Department of Mathematics University of North Carolina at Greensboro Greensboro, NC 27410-5608

Professor William A. Webb (pp. 75-82; 213-218) Department of Pure and Applied Mathematics Washington State University Pullman, WA 99164-3113

Professor Diana Wells Box 8376 University of North Dakota Grand Forks, ND 58202

xv

xvi

Professor Chizhong Zhou (pp. 369-380) Yueyang University Yueyang, Hunan 414000 P. R. China

CONTRIBUTORS TO THIS PROCEEDINGS

FOREWORD

This book contains 33 papers from among the 41 papers presented at the Eighth International Conference on Fibonacci Numbers and Their Applications which was held at the Rochester Institute of Technology, Rochester, New York, from June 22 to June 26, 1998. These papers have been selected after a careful review by well known referees in the field, and they range from elementary number theory to probability and statistics. The Fibonacci numbers and recurrence relations are their unifying bond.

It is anticipated that this book, like its seven predecessors, will be useful to research workers and graduate students interested in the Fibonacci numbers and their applications.

June 1, 1999

xvii

The Editor

F.T. Howard Mathematics and Computer Science Wake Forest University Box 7388 Reynolda Station Winston-Salem, NC USA

THE ORGANIZING COMMITTEES

LOCAL COMMITTEE

Anderson, Peter G., Chairman

Arpaya, Pasqual

Biles, John

Orr, Richard

Radziszowski, Stanislaw

Rich, Nelson

xix

INTERNATIONAL COMMITTEE

Horadam, A.F. (Australia), Co-Chair

Philippou, A.N. (Cyprus), Co-Chair

Bergum, G.E. (U.S.A.)

Filipponi, P. (Italy)

Harborth, H. (Germany)

Horibe, Y. (Japan)

Howard, F. (U.S.A.)

Johnson, M. (U.S.A.)

Kiss, P. (Hungary)

Phillips, G.M. (Scotland)

Turner, J. (New Zealand)

Waddill, M.E. (U.S.A.)

Con

fere

nce

Par

tici

pant

s

LIST OF CONTRIBUTORS TO THE CONFERENCE

AGRATINI, OCTAVIAN, "Unusual Equations in Study." *ANDO, SHIRO, (coauthor Daihachiro Sato), "On the Generalized Binomial Coefficients Defined by Strong Divisibility Sequences." *ANATASSOVA, VASSIA K., (coauthor J.C. Turner), "On Triangles and Squares Marked With Goldpoints - Studies of Golden Tiles." *ANTZOULAKOS, DEMETRIOS L., (coauthor Andreas N. Philippou), "Multivariate Pascal Polynomials of Order K with Probability Applications." *ATANASSOV, K.T., (coauthor A.G. Shannon), "Fibonacci Planes and Spaces." *BICKNELL-JOHNSON, MARJORIE, "The Smallest Positive Integer Having Fk Representations as Sums of Distinct Fibonacci Numbers." *BICKNELL-JOHNSON, MARJORIE, "The Zeckendorf-Wythoff Array Applied to Counting the Number of Representations of N as Sums of Distinct Fibonacci Numbers." *BILES, JOHN A., "Composing with Sequences: ... but is it Art?" *CARAGIU, MIHAl, (coauthor William Webb), "Invariants for Linear Recurrences." *COOPER, CURTIS, (coauthor Robert E. Kennedy), "Base 10 RATS Cycles and Arbitrarily Long Base 10 RATS Cycles." DILCHER, KARL, "Nested Squares and Evaluations of Integer Products." *ELIA, MICHELE, (coauthor Piero Filipponi), "Quintics ;e5 - 5;e - k, The Golden Section, and Square Lucas Numbers." *ERICKSEN, LARRY, "The Pascal-DeMoivre Moments and Their Generating Functions." *FIELDER, DANIEL C., (coauthor Cecil O. Alford), "Investigating Special Binary Sequences With Some Computer Help." FIELDER, DANIEL C., (coauthor Marjorie Bicknell-Johnson), "The 1st 330 terms of Sequence AOI3583." *FILIPPONI, PIERO, (coauthor Alwyn F. Horadam), "Integration Sequences of Jacobsthal and Jacobsthal-Lucas Polynomials." *FREITAG, HERTA T., (coauthors Marjorie Bicknell-Johnson and George M. Phillips), "A Property of the Unit Digits of Recursive Sequences." *FREITAG, HERTA T., (coauthor Daniel C. Fielder), "On General Divisibility of Sums of Integral Powers of the Golden Ratio." *FREITAG, HERTA T., (coauthor George M. Phillips), "Sylvester's Algorithm and Fibonacci Numbers." *GROSSMAN, GEORGE W., (coauthor Sivaram K. Narayan), "On the Characteristic Polynomial of the j-th Order Fibonacci Sequence." *HORADAM, A.F., "Quasi Morgan-Voyce Polynomials and Pell Convolutions." *HORIBE, YASUICHI, "On an Asymptotic Maximality of the Fibonacci Tree." *HOWARD, F.T., "Generalizations of a Fibonacci Identity." *KIMBALL, WILLIAM A., (coauthor William A. Webb), "Some Generalizations of Wolstenholme's Theorem."

*The asterisk indicates that the paper is included in this book.

xxi

xxii CONTRIBUTORS TO THE CONFERENCE

*KIMBERLING, CLARK, "Card Sorting Related to Fibonacci Numbers." *LEE, JACK, "On the Inhomogeneous Geometric Line-Sequence." *LUCA, FLORIAN, "Fibonacci Numbers of the Form k2 + k + 2." *MELHAM, R.S., "On Certain Polynomials of Even Subscripted Lucas Numbers." MORGAN, MARK D., "The Distribution of Second Order Linear Recurrence Sequences Mod 2m." *MULLER, SIGUNA, "On the Rank of Appearance of Lucas Sequences." *RABINOWITZ, STANLEY, "Algorithmic Simplification of Reciprocal Sums." RABINOWITZ, STANLEY, "Proving identities by Computer." *ROTKIEWICZ, ANDRZEJ, "Solved and Unsolved Problems on Pseudoprime Numbers and Their Generalizations." SAN CHIS, LAURA, "On the Occurrence of Certain Fibonacci Numbers in the Zeckendorf Decomposition of nF n." *SHANNON, A.G., (coauthor A.F. Horadam), "Some Relationships Among Vieta, Morgan­Voyce and Jacobsthal Polynomials." *SOMER, LAWRENCE, "Special Multipliers of Lucas Sequences Modulo pr." *SZYBIST, JOHN, (coauthor Peter G. Anderson), "Digital Half toning Using Error Diffusion and Linear Pixel Shuffiing." TAKAHASHI, TOMAKAZU, "Forcasting Method for Stock Prices by Using Elliott Wave Principle and Neural Network." *TURNER, JOHN C., "On Vector Sequence Recurrence Equations in Fibonacci Vector Geometry." *ZHOU, CHIZHONG, "Constructing Identities Involving Kth-order F-L Numbers by Using the Characteristic Polynomial."

INTRODUCTION

The Fibonacci numbers

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... ,

were first mentioned in 1202 in the Liber Abaci, a book written by Leonardo of Pisa to introduce the Hindu-Arabic numeral system to western Europe. Leonardo, perhaps the greatest mathematician of the Middle Ages, wrote under the name of Fibonacci -- a contraction of "filius Bonacci" (son of Bonacci). In Liber Abaci the numbers appeared in the famous rabbit problem, but they were not called "Fibonacci numbers" until the nineteenth century, when the French mathematician Edouard Lucas used that term. Lucas studied the Fibonacci numbers extensively, and the simple generalization

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ... ,

bears his name.

The rich and interesting history of the Fibonacci numbers in the eighteenth and nineteenth centuries can be found in 1. E. Dickson's History of the Theory of Numbers, volume 1. During the twentieth century, interest in Fibonacci numbers and their applications rose rapidly. In 1961 the Soviet mathematician N. Vorobyov published Fibonacci Numbers, and Verner E. Hoggatt, Jr., followed in 1969 with his Fibonacci and Lucas Numbers. Meanwhile, in 1963, Hoggatt and his associates founded The Fibonacci Association and began publishing The Fibonacci Quarterly. They also organized a Fibonacci Conference in California, U.S.A., each year for almost sixteen years until 1979.

In 1984, the First International Conference on Fibonacci Numbers and Their Applications was held in Patras, Greece, and the proceedings from that conference were published. It was anticipated at that time that this conference would be the first of a series of international conferences on the subject to be held every two or three years in different countries. With this intention as a motivating force, The Second, Third, Fourth, Fifth, Sixth and Seventh International Conference on Fibonacci Numbers and Their Applications were respectively held in alternate years at San Jose, California; Pisa, Italy; Winston-Salem, North Carolina; St. Andrews, Scotland; Pullman, Washington and Graz, Austria. The proceedings from these seven conferences have also been published. Because of the continuous success of the preceding seven conferences, The Eighth International Conference on Fibonacci Numbers and Their Applications was held at Rochester, New York, June 22-26, 1998, and a Ninth Conference is scheduled in July 2000 in Luxembourg.

It is impossible to overemphasize the importance and relevance of the Fibonacci numbers to the mathematical and physical sciences as well as other areas of study. The Fibonacci numbers appear in almost every branch of mathematics, including number theory, combinatorics, differential equations, probability, statistics, numerical analysis, and linear algebra. They also occur in physics, biology, chemistry, and electrical engineering.

xxiii

xxiv INTRODUCTION

It is believed that the contents of this book, like its predecessors, will prove useful to everyone interested in this important branch of mathematics and that this material may lead to additional results on Fibonacci numbers both in mathematics and in their applications to science and engineering.

The editor would like to acknowledge The Fibonacci Association and Rochester Institute of Technology for their financial and other assistance in making the conference a success. He would also like to thank Gerald Bergum, Marjorie Bicknell-Johnson and Calvin Long for their advice, assistance and encouragement. Finally, the editor thanks the technical typist, Patricia Solsaa, for her excellent work.

The Editor